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c ○ World Scientific Publishing Company QUENCHING LORENZIAN CHAOS RAYMOND HIDE.pdf

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c ○ World Scientific Publishing Company QUENCHING LORENZIAN CHAOS RAYMOND HIDE

September 2, 2004 14:10 01090 International Journal of Bifurcation and Chaos, Vol. 14, No. 8 (2004) 2875–2884 c? World Scientific Publishing Company QUENCHING LORENZIAN CHAOS RAYMOND HIDE Department of Mathematics, Imperial College London, London SW7 2AZ, UK PATRICK E. McSHARRY Mathematical Institute, Oxford University, Oxford OX1 3LB, UK CHRISTOPHER C. FINLAY? and GUY D. PESKETT Department of Physics, Oxford University, Oxford OX1 3PU, UK Received December 16, 2002 How fluctuations can be eliminated or attenuated is a matter of general interest in the study of steadily-forced dissipative nonlinear dynamical systems. Here, we extend previous work on “nonlinear quenching” [Hide, 1997] by investigating the phenomenon in systems governed by the novel autonomous set of nonlinear ordinary differential equations (ODE’s) x? = ay ? ax, y? = ?xzq + bx ? y and z? = xyq ? cz (where (x, y, z) are time(t)-dependent dimensionless variables and x? = dx/dt, etc.) in representative cases when q, the “quenching function”, satisfies q = 1 ? e + ey with 0 ≤ e ≤ 1. Control parameter space based on a, b and c can be divided into two “regions”, an S-region where the persistent solutions that remain after initial transients have died away are steady, and an F-region where persistent solutions fluctuate indefinitely. The “Hopf boundary” between the two regions is located where b = bH(a, c; e) (say), with the much studied point (a, b, c) = (10, 28, 8/3), where the persistent “Lorenzian” chaos that arises in the case when e = 0 was first found lying close to b = bH(a, c; 0). As e increases from zero the S-region expands in total “volume” at the expense of F-region, which disappears altogether when e = 1 leaving persistent solutions that are steady throughout the entire parameter space. Keywords : Nonlinear quenching; chaos; nonlinear circuits. 1. Introduction Many investigations of chaos have been concerned with steadily-forced nonlinear dissipative physical systems governed by autonomous set